Introduction to Shape Analysis

The starting point of my thesis is a simple observation: most objects commonly used in computer science, such as arrays, trees or graphs, can be separated into two parts a \shape" part, describing the \structure" of the object, and a list of data. In this way we can split an array into the list of dimensions and the list of entries, a tree with labeled nodes into the underlying unlabeled tree and a list of labels, a graph with weighted edges into an unlabeled graph and a list of weights (in fact, in case of graphs we need an implicit ordering of edges as well). A closer look at this idea reveals that, for example, not every tuple consisting of a list of dimensions and a list of entries gives us an array the product of dimensions has to be equal to the length of the list of entries. The same condition holds for trees the number of nodes has to be the same as the length of the list of labels. So for every \shape" (an unlabeled tree, a list of dimensions) we have its \arity" the number of data that ts into it. We can think about shape as a structure with some holes in it the number of these holes tells us how many pieces of data we have to put in to get a complete object. Categorically, this situation is captured by a pullback